The advantage of this method would be that it can be computed analytically for an arbitrary system and is only based on the computation of the inverse of the matrix \(\left(\rho \otimes +\otimes \rho ^{*} \right)\) . For high-dimensional systems, efficient methods such as Cholesky decomposition may be required [1]}. It is worth noting that the above formula is only applicable if the density matrix \(\rho \)  has full rank and thus is invertible. If \(\rho \)  is not invertible, Šafránek gives another formal solution of quantum Fisher information matrix by using the invertible matrix \(\rho _{s}\equiv \left(1-s\right)\rho +\frac{s}{d}\)  corresponding to \(\rho \)  such that the equation (REF ) becomes
\({\cal F}_{\theta _{\mu }\theta _{\nu }}=2\lim _{s\rightarrow 0}{\rm vec}\left[\partial _{\theta _{\mu }}\rho _{s}\right]^{\dagger }\left(\rho _{s}\otimes +\otimes \rho _{s}^{*} \right)^{-1}{\rm vec}\left[\partial _{\theta _{\nu }}\rho _{s} \right].\) 
